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Theorem cnextf 19479
Description: Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
Hypotheses
Ref Expression
cnextf.1  |-  C  = 
U. J
cnextf.2  |-  B  = 
U. K
cnextf.3  |-  ( ph  ->  J  e.  Top )
cnextf.4  |-  ( ph  ->  K  e.  Haus )
cnextf.5  |-  ( ph  ->  F : A --> B )
cnextf.a  |-  ( ph  ->  A  C_  C )
cnextf.6  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  C )
cnextf.7  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
Assertion
Ref Expression
cnextf  |-  ( ph  ->  ( ( JCnExt K
) `  F ) : C --> B )
Distinct variable groups:    x, A    x, B    x, C    x, F    x, J    x, K    ph, x

Proof of Theorem cnextf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cnextf.3 . . . 4  |-  ( ph  ->  J  e.  Top )
2 cnextf.4 . . . 4  |-  ( ph  ->  K  e.  Haus )
3 cnextf.5 . . . 4  |-  ( ph  ->  F : A --> B )
4 cnextf.a . . . 4  |-  ( ph  ->  A  C_  C )
5 cnextf.1 . . . . 5  |-  C  = 
U. J
6 cnextf.2 . . . . 5  |-  B  = 
U. K
75, 6cnextfun 19477 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Haus )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  Fun  ( ( JCnExt K
) `  F )
)
81, 2, 3, 4, 7syl22anc 1212 . . 3  |-  ( ph  ->  Fun  ( ( JCnExt
K ) `  F
) )
9 simpl 454 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  ph )
10 cnextf.6 . . . . . . . . 9  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  C )
1110eleq2d 2500 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  C ) )
1211biimpar 482 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  ( ( cls `  J
) `  A )
)
13 cnextf.7 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
14 n0 3634 . . . . . . . 8  |-  ( ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/)  <->  E. y 
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F ) )
1513, 14sylib 196 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  E. y 
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F ) )
16 haustop 18776 . . . . . . . . . . . . . 14  |-  ( K  e.  Haus  ->  K  e. 
Top )
172, 16syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  Top )
185, 6cnextfval 19475 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  (
( JCnExt K ) `
 F )  = 
U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
191, 17, 3, 4, 18syl22anc 1212 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( JCnExt K
) `  F )  =  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2019eleq2d 2500 . . . . . . . . . . 11  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
)  <->  <. x ,  y
>.  e.  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
21 opeliunxp 4877 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2220, 21syl6bb 261 . . . . . . . . . 10  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
)  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
2322exbidv 1679 . . . . . . . . 9  |-  ( ph  ->  ( E. y <.
x ,  y >.  e.  ( ( JCnExt K
) `  F )  <->  E. y ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
24 19.42v 1922 . . . . . . . . 9  |-  ( E. y ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2523, 24syl6bb 261 . . . . . . . 8  |-  ( ph  ->  ( E. y <.
x ,  y >.  e.  ( ( JCnExt K
) `  F )  <->  ( x  e.  ( ( cls `  J ) `
 A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
2625biimpar 482 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( cls `  J
) `  A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )  ->  E. y <. x ,  y >.  e.  ( ( JCnExt K ) `
 F ) )
279, 12, 15, 26syl12anc 1209 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)
2825simprbda 618 . . . . . . 7  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  x  e.  ( ( cls `  J
) `  A )
)
2911adantr 462 . . . . . . 7  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  C ) )
3028, 29mpbid 210 . . . . . 6  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  x  e.  C )
3127, 30impbida 821 . . . . 5  |-  ( ph  ->  ( x  e.  C  <->  E. y <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
) ) )
3231abbi2dv 2548 . . . 4  |-  ( ph  ->  C  =  { x  |  E. y <. x ,  y >.  e.  ( ( JCnExt K ) `
 F ) } )
33 dfdm3 5014 . . . 4  |-  dom  (
( JCnExt K ) `
 F )  =  { x  |  E. y <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
) }
3432, 33syl6reqr 2484 . . 3  |-  ( ph  ->  dom  ( ( JCnExt
K ) `  F
)  =  C )
35 df-fn 5409 . . 3  |-  ( ( ( JCnExt K ) `
 F )  Fn  C  <->  ( Fun  (
( JCnExt K ) `
 F )  /\  dom  ( ( JCnExt K
) `  F )  =  C ) )
368, 34, 35sylanbrc 657 . 2  |-  ( ph  ->  ( ( JCnExt K
) `  F )  Fn  C )
3719rneqd 5054 . . 3  |-  ( ph  ->  ran  ( ( JCnExt
K ) `  F
)  =  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
38 rniun 5235 . . . 4  |-  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  = 
U_ x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
39 vex 2965 . . . . . . . . 9  |-  x  e. 
_V
4039snnz 3981 . . . . . . . 8  |-  { x }  =/=  (/)
41 rnxp 5256 . . . . . . . 8  |-  ( { x }  =/=  (/)  ->  ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
4240, 41ax-mp 5 . . . . . . 7  |-  ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )
4311biimpa 481 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  x  e.  C )
446toptopon 18379 . . . . . . . . . . 11  |-  ( K  e.  Top  <->  K  e.  (TopOn `  B ) )
4517, 44sylib 196 . . . . . . . . . 10  |-  ( ph  ->  K  e.  (TopOn `  B ) )
4645adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  K  e.  (TopOn `  B )
)
475toptopon 18379 . . . . . . . . . . . 12  |-  ( J  e.  Top  <->  J  e.  (TopOn `  C ) )
481, 47sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  C ) )
4948adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  J  e.  (TopOn `  C )
)
504adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  A  C_  C )
51 simpr 458 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  C )
52 trnei 19306 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  C )  /\  A  C_  C  /\  x  e.  C )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
5352biimpa 481 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  C )  /\  A  C_  C  /\  x  e.  C )  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
5449, 50, 51, 12, 53syl31anc 1214 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
553adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  F : A --> B )
56 flfelbas 19408 . . . . . . . . . . 11  |-  ( ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  /\  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  -> 
y  e.  B )
5756ex 434 . . . . . . . . . 10  |-  ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  ->  (
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F )  ->  y  e.  B ) )
5857ssrdv 3350 . . . . . . . . 9  |-  ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  C_  B
)
5946, 54, 55, 58syl3anc 1211 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  C_  B
)
6043, 59syldan 467 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ( ( K  fLimf  ( ( ( nei `  J ) `
 { x }
)t 
A ) ) `  F )  C_  B
)
6142, 60syl5eqss 3388 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ran  ( { x }  X.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6261ralrimiva 2789 . . . . 5  |-  ( ph  ->  A. x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
63 iunss 4199 . . . . 5  |-  ( U_ x  e.  ( ( cls `  J ) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B 
<-> 
A. x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6462, 63sylibr 212 . . . 4  |-  ( ph  ->  U_ x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6538, 64syl5eqss 3388 . . 3  |-  ( ph  ->  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6637, 65eqsstrd 3378 . 2  |-  ( ph  ->  ran  ( ( JCnExt
K ) `  F
)  C_  B )
67 df-f 5410 . 2  |-  ( ( ( JCnExt K ) `
 F ) : C --> B  <->  ( (
( JCnExt K ) `
 F )  Fn  C  /\  ran  (
( JCnExt K ) `
 F )  C_  B ) )
6836, 66, 67sylanbrc 657 1  |-  ( ph  ->  ( ( JCnExt K
) `  F ) : C --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362   E.wex 1589    e. wcel 1755   {cab 2419    =/= wne 2596   A.wral 2705    C_ wss 3316   (/)c0 3625   {csn 3865   <.cop 3871   U.cuni 4079   U_ciun 4159    X. cxp 4825   dom cdm 4827   ran crn 4828   Fun wfun 5400    Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080   ↾t crest 14341   Topctop 18339  TopOnctopon 18340   clsccl 18463   neicnei 18542   Hauscha 18753   Filcfil 19259    fLimf cflf 19349  CnExtccnext 19472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-map 7204  df-pm 7205  df-rest 14343  df-fbas 17657  df-fg 17658  df-top 18344  df-topon 18347  df-cld 18464  df-ntr 18465  df-cls 18466  df-nei 18543  df-haus 18760  df-fil 19260  df-fm 19352  df-flim 19353  df-flf 19354  df-cnext 19473
This theorem is referenced by:  cnextcn  19480  cnextfres  19481
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